Exponents & Powers

Exponents and powers are fundamental concepts in algebra and are frequently tested on the GMAT. Understanding the rules and properties of exponents is essential for simplifying expressions and solving equations efficiently.

Basic Definitions

An exponent represents the number of times a base is multiplied by itself. It is written as:

$a^n$

where:

• $a$ is the base,
• $n$ is the exponent or power.

For example, $2^3 = 2 \times 2 \times 2 = 8$.

Rules of Exponents

Here are the key rules of exponents that are frequently used in GMAT problems:

• Product Rule: $a^m \times a^n = a^{m+n}$
• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$ (for $a \neq 0$)
• Power Rule: $(a^m)^n = a^{m \times n}$
• Zero Exponent Rule: $a^0 = 1$ (for $a \neq 0$)
• Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$ (for $a \neq 0$)
• Fractional Exponent Rule: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$

Special Cases

There are some special cases and properties of exponents that are useful to know:

• Base of 1: $1^n = 1$ for any exponent $n$.
• Base of 0: $0^n = 0$ for any positive exponent $n$. Note that $0^0$ is undefined.
• Negative Base: If the base is negative, the result depends on whether the exponent is odd or even:
  • Odd exponent: $(-a)^n = -a^n$ (e.g., $(-2)^3 = -8$)
  • Even exponent: $(-a)^n = a^n$ (e.g., $(-2)^2 = 4$)

Applications in GMAT Problems

Exponents and powers are commonly tested in GMAT problems involving:

• Simplifying algebraic expressions.
• Solving exponential equations.
• Comparing the magnitude of numbers with exponents.
• Solving word problems involving exponential growth or decay.

Tips for Solving Exponent Problems

• Memorize the rules of exponents and practice applying them.
• Look for opportunities to simplify expressions using exponent rules.
• Be cautious with negative bases and fractional exponents.
• Use estimation to compare the magnitude of numbers with exponents.

Mastering exponents and powers will help you tackle a wide range of GMAT problems efficiently and improve your quantitative score.